Vector Algebra 12.1 12.2 28 August 2013
What is a Vector? A vector (denoted or v) is a mathematical object possessing both: direction and magnitude also called length (denoted ). Vectors are often represented by arrows.
Three Examples of Vectors: Example 1: Displacement Vectors. The displacement vector PQ measures the displacement of an object that moves along along a line segment from a point P to a point Q. For a displacement vector: The magnitude PQ is the distance from P to Q (the distance traveled by the object). The direction of PQ is the direction of displacement (the direction along which the object moves).
Three Examples of Vectors: Example 2: Velocity Vectors. The velocity of a moving object can be represented by a vector. The magnitude is the speed. The direction of is the direction in which the object is moving at that instant.
Three Examples of Vectors: Example 3: Force Vectors. A force F can be measured by a vector. The magnitude F is the magnitude of the force. The direction of F is the direction in which the force is acting.
Cartesian (Rectangular) Coordinates. Fix an origin O: In R 2 (the plane), points can be represented as ordered pairs of numbers: P = (x 0, y 0 ) In R 3 (3-space), points can be represented as ordered triples of numbers: P = (x 0, y 0, z 0 ).
Vectors in Component Form. Coordinates allow us to write vectors in component form. If P = (x 1, y 1, z 1 ) and Q = (x 2, y 2, z 2 ) then PQ = x 2 x 1, y 2 y 1, z 2 z 1.
Magnitude. If = v 1, v 2, = v 2 1 + v 2 2 If = v 1, v 2, v 3, = v 2 1 + v 2 2 + v 2 3 The magnitude of a vector is the distance between its endpoints. The distance formula is derived from the Pythagorean Theorem. Magnitude is a scalar quantity.
Vector Operations: Vector Addition. = v 1, v 2, v 3 = w 1, w 2, w 3 Algebraic: add component-wise: + = v 1 + w 1, v 2 + w 2, v 3 + w 3. Geometric: Tip-to-tail addition, or Parallelogram Law.
Vector Operations: Vector Addition. = v 1, v 2, v 3 = w 1, w 2, w 3 Algebraic: add component-wise: + = v 1 + w 1, v 2 + w 2, v 3 + w 3. Geometric: Tip-to-tail addition, or Parallelogram Law.
Vector Operations: Vector Addition. = v 1, v 2, v 3 = w 1, w 2, w 3 Algebraic: add component-wise: + = v 1 + w 1, v 2 + w 2, v 3 + w 3. Geometric: Tip-to-tail addition, or Parallelogram Law. +
Vector Operations: Vector Addition. = v 1, v 2, v 3 = w 1, w 2, w 3 Algebraic: add component-wise: + = v 1 + w 1, v 2 + w 2, v 3 + w 3. Geometric: Tip-to-tail addition, or Parallelogram Law. +
Vector Operations: Vector Addition. = v 1, v 2, v 3 = w 1, w 2, w 3 Algebraic: add component-wise: + = v 1 + w 1, v 2 + w 2, v 3 + w 3. Geometric: Tip-to-tail addition, or Parallelogram Law. +
Vector Operations: Vector Addition. = v 1, v 2, v 3 = w 1, w 2, w 3 Algebraic: add component-wise: + = v 1 + w 1, v 2 + w 2, v 3 + w 3. Geometric: Tip-to-tail addition, or Parallelogram Law. + +
Vector Operations: Scalar Multiplication. = v 1, v 2, v 3 c R (c is a scalar) Algebraic: multiply component-wise: c = cv 1, cv 2, cv 3 Geometric: stretch, shrink, or change the direction of the vector. Scalar times vector gives a vector.
Vector Operations: Vector Subtraction. Algebraic: = v 1, v 2, v 3 = w 1, w 2, w 3 Geometric: = + ( 1) = v 1 w 1, v 2 w 2, v 3 w 3. is the vector from the tip of to the tip of.
Vector Operations: Vector Subtraction. Algebraic: = v 1, v 2, v 3 = w 1, w 2, w 3 Geometric: = + ( 1) = v 1 w 1, v 2 w 2, v 3 w 3. is the vector from the tip of to the tip of.
Vector Operations: Vector Subtraction. Algebraic: = v 1, v 2, v 3 = w 1, w 2, w 3 = + ( 1) = v 1 w 1, v 2 w 2, v 3 w 3. Geometric: is the vector from the tip of to the tip of.
Vector Operations: Vector Subtraction. Algebraic: = v 1, v 2, v 3 = w 1, w 2, w 3 = + ( 1) = v 1 w 1, v 2 w 2, v 3 w 3. Geometric: is the vector from the tip of to the tip of.
Vector Operations: Vector Subtraction. Algebraic: = v 1, v 2, v 3 = w 1, w 2, w 3 = + ( 1) = v 1 w 1, v 2 w 2, v 3 w 3. Geometric: is the vector from the tip of to the tip of.
Unit Vectors. is called a unit vector if = 1. If 0, then the unit vector in the direction of is: e = 1. Sometimes e is denoted by ˆv.
Clicker Question: Unit Vectors Which of the following vectors is not a unit vector? A. 0, 1 B. 1, 0 C. 1, 1 D. 1/ 2, 1/ 2 E. I don t understand the question. receiver channel: 41 session ID: bsumath275
Clicker Question: Unit Vectors Which of the following vectors is not a unit vector? A. 0, 1 B. 1, 0 C. 1, 1 D. 1/ 2, 1/ 2 E. I don t understand the question. receiver channel: 41 session ID: bsumath275
Basis Vectors: î, ĵ, ˆk In R 2 : î = 1, 0 ĵ = 0, 1 In R 3 : î = 1, 0, 0 ĵ = 0, 1, 0 ˆk = 0, 0, 1 These are unit vectors pointing in the positive direction along the coordinate axes.
Vector Representations = v 1, v 2, v 3 Representing in terms of the basis vectors î, ĵ, ˆk: = v 1 î + v 2 ĵ + v 3 ˆk. Representing (if 0) in terms of direction and magnitude: = ê.
Angle with x-axis θ v sin θ v cos θ If in R 2 makes an angle of θ with the positive x-axis then = v cos θ, v sin θ.
Linear combinations A linear combination of and is a vector r + s, where r, s are scalars. Example: For = 5, 2, = 2, 1, what is the linear combination + 3?
Linear combinations A linear combination of and is a vector r + s, where r, s are scalars. Example: For = 5, 2, = 2, 1, what is the linear combination + 3? 3
Linear combinations A linear combination of and is a vector r + s, where r, s are scalars. Example: For = 5, 2, = 2, 1, what is the linear combination + 3? 3
Linear combinations A linear combination of and is a vector r + s, where r, s are scalars. Example: For = 5, 2, = 2, 1, what is the linear combination + 3? 3 + 3
Linear combinations = 5, 2 = 2, 1 Example: Can 8, 5 be written as a linear combination of and? Answer:
Linear combinations = 5, 2 = 2, 1 Example: Can 8, 5 be written as a linear combination of and? Answer: Try to solve 8, 5 = r 5, 2 + s 2, 1, so 5r + 2s = 8 2r s = 5
Linear combinations = 5, 2 = 2, 1 Example: Can 8, 5 be written as a linear combination of and? Answer: Try to solve 8, 5 = r 5, 2 + s 2, 1, so 5r + 2s = 8 2r s = 5 Use elimination, or substitution to solve: r = 2, s = 1. 8, 5 = 2.
Clicker Question: Linear combinations In this picture, the unlabeled vector is closest to which of the following: A. + B. C. + 2 D. 2 + E. I don t understand the question. receiver channel: 41 session ID: bsumath275
Clicker Question: Linear combinations In this picture, the unlabeled vector is closest to which of the following: A. + B. C. + 2 D. 2 + E. I don t understand the question. receiver channel: 41 session ID: bsumath275